You want to modify the functions fn(x)=xn (n = 1, 2, 3,
...) in the following way: Outside the interval [-1,1] the functions should
remain completely unchanged. At zero, the new functions should be 1
(instead of 0). And the most important requirement: The functions should
remain continuously differentiable infinitely many times, everywhere.

You can achieve this by adding a "hump" function to fn.
But not any function will do: For instance, you cannot add a Gaussian exp(-x2),
because that would change fn outside the interval [-1,1]. You also
cannot add a sine hump, (0.5 + 0.5*sin(π x)) for x in [-1,1] and 0 otherwise, because the
second derivative would become discontinuous at -1 and 1. Find a hump function
which will do the trick.